Dispersion in Maths

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    1 Arithmetic mean:

    (i) For ungrouped data (individual series) x =

    (ii) For grouped data (continuous series)

    (a) Direct method x = where xi, I = 1 .. n be n observations and fi be their corresponding

    frequencies

    (b) short cut method : x = A + fidi / fi) where A = assumed mean, d i = xi - A = deviation for each term

    2 Properties of A.M.

    (i) In a statistical data the sum of the deviation of items form A.M. is alwalys zero.

    (ii) If each of the n given observation be doubled, then their mean is doubled

    (iii) Ifx is the mean of x1, x2, , xn . the mean of ax1, x2, .. , axn is a x where a is any number

    different form zero(iv) Arithmetic mean is independent of origin i.e. it is x effected by any change in origin.

    3 Geometric mean:

    (i) For ungrouped data G.M. = (x1 x2 x3.. xn)1/n or

    G.M. = antilog

    (ii) For grouped data G.M. = (xf1

    1 xf22.. x

    fnn)

    1/N, where N = i=1

    nfi

    4 Harmonic mean Harmonic mean is reciprocal of arithmetic mean of reciprocals.

    (i) For ungrouped data H.M. =

    (ii) For grouped data H.M. =

    5 Relation between A.M., G.M and H.M.

    A.M. G.M. H.M.

    Equation holds only when all the observations in the series are same

    6 Msdian:

    (a) Individual series (ungrouped data) : If data is raw, arrange in ascending or descending order and n be

    the no. of observations . If n is odd, Median = value of ((n+1) / 2))th observation If n is even, median =

    (1/2) [value of (n/2)th + value of ((n/2) + 1)th] observation.

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    (b) Discrete series: First find cumulative frequencies of the variables arranged in ascending or

    descending order and Mediain = {(n+1) / 2}th

    observation, where n is cumulative frequency.

    (c) Continuous distribution (grouped data)

    (i) For series in ascending order

    Median = e +

    Where

    e = Lower limit of the median class.

    f = Frequency of the median class.

    N = Sum of the all frequencies.

    i = The width of the median class.

    C = Cumulative frequency of the class preceding to median class.

    (ii) For series in descending order Median = u -

    Where u = upper limit of median class.

    7 Mode:

    (i) For individual series: In the case of individual series, the value which is repeated maximum number of

    times is the mode of the series.

    (ii) For discrete frequency distribution series: In the case of discrete frequency distribution, mode is the

    value of the variate corresponding to the maximum frequency.

    (iii) For continuous frequency distribution : first find the model class i.e. the class which has maximum

    frequency for continuous series

    Where

    e1 = Lower limit of the model class.

    f1 = Frequency of the model class.

    f0 = Frequency of the class preceding mode class.

    f2 = Frequency of the class succeeding model class.

    i= Size of the model class.

    8 Relation between mean, mode & median:

    (i) In symmetrical distribution : mean = mode = median

    (ii) In Moderately symmetrical distribution : mode = 3 median 2 mean

    Measure of Dispersion:

    The degree to which numerical data tend to spread about an average value is called variation or

    dispersion popular methods of measure of dispersion.

    (a) Individual series (ungrouped data)

    1 Mean deviationMean deviation = (|x-S| / n)

    Where n = number of terms, S = deviation of variety form mean mode , median

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    (b) Continuous series (grouped data)

    Note:

    Mean deviation is the least when measured from the median.

    2 Standerd Deviation :

    S.D. () is the squere root of the arithmetic mean of the squares of the deviations of the terms from

    their A.M.

    (a) for individual series (ungrouped data)

    where x = Arithmetic mean of the series N = Total frequency

    (b) For continuous series (grouped data)

    (i) Direct method =

    Where

    x = Arithmetic mean of series

    X1 = mid value of the class

    f1 = Frequency of the corresponding x1

    N = f = Total frequency

    (ii) Short cut method

    Where

    d = x - A = Derivation from assumed mean A

    f = Frequency of item (term)

    N = f = Total frequency.

    Variance Square of standard direction

    i.e. variance = (S.D.)2 = ()2

    Coefficient of variance = Coefficient of S.D. x 100 = ( / x) x 100